A010572 -id:A010572 - cp's OEIS frontend

A010571 High temperature expansion of -u/J in odd powers of v = tanh(J/kT), where u is energy per site of the spin-1/2 Ising model on cubic lattice with nearest-neighbor interaction J at temperature T.

3, 12, 120, 1368, 18300, 268728, 4179852, 67767744, 1133826324, 19443072084, 340085761968, 6046276240668, 108970501777080, 1986820814551056, 36587507853481908, 679619087721892176, 12720247240214281860, 239685390231729125004, 4543441582487318876664

Offset: 1

N. J. A. Sloane

M. E. Fisher and D. S. Gaunt, Ising model and self-avoiding walks on hypercubical lattices and high density expansions, Phys. Rev., 133 (1964), A224-A239. See Table II for correct a(1)-a(6) but incorrect a(7).
M. E. Fisher and M. F. Sykes, Antiferromagnetic susceptibilities of the simple cubic and body-centered cubic Ising lattices, Physica, 28 (1962), 939-956.

Cf. A001393, A002908, A007239, A010572-A010574, A047711, A047712.

Sum_{n>=1} a(n) * v^(2*n-1) = v*q/2 + (1-v^2) * f'(v) / f(v), where f(v) = Sum_{n>=0} A001393(n) * v^(2*n) and q = 6 is the number of nearest neighbors. - Andrey Zabolotskiy, Feb 14 2022

New name from Andrey Zabolotskiy, Jan 14 2019

a(7) corrected and more terms added by Andrey Zabolotskiy, Feb 14 2022

A002908 High temperature expansion of -u/J in odd powers of v = tanh(J/kT), where u is energy per site of the spin-1/2 Ising model on square lattice with nearest-neighbor interaction J at temperature T.

Original entry on oeis.org

2, 4, 8, 24, 84, 328, 1372, 6024, 27412, 128228, 613160, 2985116, 14751592, 73825416, 373488764, 1907334616, 9820757380, 50934592820, 265877371160, 1395907472968, 7366966846564, 39062802311672, 208015460898924, 1112050252939612, 5966352507546872

Offset: 1

N. J. A. Sloane, Simon Plouffe

Previous name was: Energy function for square lattice.

C. Domb, Ising model, in Phase Transitions and Critical Phenomena, vol. 3, ed. C. Domb and M. S. Green, Academic Press, 1974; p. 386.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

C. Domb, Ising model, Phase Transitions and Critical Phenomena 3 (1974), 257, 380-381, 384-387, 390-391, 412-423. (Annotated scanned copy)
M. E. Fisher and D. S. Gaunt, Ising model and self-avoiding walks on hypercubical lattices and high density expansions, Phys. Rev. 133 (1964) A224-A239.
Lars Onsager, Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition, Phys. Rev. 65, 117 (1944).
M. F. Sykes and M. E. Fisher, Antiferromagnetic susceptibility of the plane square and honeycomb Ising lattices, Physica, 28 (1962), 919-938.

Cf. A002906-A002930, A010571, A010572, A010573, A010574.

series((1+v^2)*(1-(2/Pi)*(1-6*v^2+v^4)*EllipticK(4*v*(1-v^2)/(1+v^2)^2)/(1+v^2)^2)/2*v,v,50); # Sean A. Irvine, Nov 26 2017

u[h_]:=Coth[2h](1+(2/Pi)(2Tanh[2h]^2-1)EllipticK[(2Sinh[2h]/Cosh[2h]^2)^2]);
Table[SeriesCoefficient[u[ArcTanh[v]],{v,0,2n-1}],{n,10}]
(* Andrey Zabolotskiy, Sep 12 2017; see Onsager's eq. (116) *)
Rest[CoefficientList[Series[(1+x)/2 - (1 - 6*x + x^2)*EllipticK[(16*(-1 + x)^2*x)/(1 + x)^4] / (Pi*(1+x)), {x, 0, 25}], x]] (* Vaclav Kotesovec, Apr 27 2024 *)

a(n) ~ 2 * (1 + sqrt(2))^(2*n-1) / (Pi * n^2). - Vaclav Kotesovec, Apr 27 2024

More terms and new name from Andrey Zabolotskiy, Oct 19 2017

A010557 Fourth-field derivative of Ising model free energy for 4-d cubic lattice.

Original entry on oeis.org

1, 32, 584, 8288, 101240, 1121120, 11570360, 113293088, 1064631032, 9681082144, 85688330696, 741562925664, 6296196525768, 52589092312288, 433044168426616, 3521747918221984, 28326976016327032, 225625290109912096, 1781402824552864712, 13954143265951219296, 108525895871787179576

Offset: 0

N. J. A. Sloane

nonn

P. Butera and M. Pernici, High-temperature expansions of the higher susceptibilities for the Ising model in general dimension d, Phys. Rev. E 86, 011139 (2012); arXiv:1209.3592 [hep-lat], 2012.
D. S. Gaunt, M. F. Sykes and S. McKenzie, Susceptibility and fourth-field derivative of the spin-1/2 Ising model for T > T_c and d = 4, J. Phys. A 12 (1979), 871-877.

Cf. A010047, A010556, A010572.

Name clarified, a(17)-a(20) using Butera & Pernici's formulas added by Andrey Zabolotskiy, Nov 25 2024

A010563 High-temperature expansion for Ising model spin-spin correlation function on 4-d cubic lattice.

Original entry on oeis.org

1, 6, 108, 2628, 72638, 2200108, 71072290, 2405434612

Offset: 0

N. J. A. Sloane

M. A. Moore, Critical behavior of the four-dimensional Ising ferromagnet and the breakdown of scaling, Phys. Rev. B 1 (1970), 2238-2240.

Cf. A010572, A030044 (partition function), A010557 (fourth-field derivative of free energy); A010564 ("HBCC" lattice), A010565 (D_4 lattice).

a(n) = A010572(n) / 4. - Andrey Zabolotskiy, Nov 19 2024

Name clarified and terms a(6)-a(7) using data from A010572 added by Andrey Zabolotskiy, Nov 25 2024

cp's OEIS Frontend

A010571 High temperature expansion of -u/J in odd powers of v = tanh(J/kT), where u is energy per site of the spin-1/2 Ising model on cubic lattice with nearest-neighbor interaction J at temperature T.

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A002908 High temperature expansion of -u/J in odd powers of v = tanh(J/kT), where u is energy per site of the spin-1/2 Ising model on square lattice with nearest-neighbor interaction J at temperature T.

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A010557 Fourth-field derivative of Ising model free energy for 4-d cubic lattice.

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A010563 High-temperature expansion for Ising model spin-spin correlation function on 4-d cubic lattice.

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